Question

solve for t. separate multiple solutions with a comma. if there are no solutions, enter none. \\(\frac{t}{t - 42} = -\frac{2}{5}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{t}{t - 42}=-\frac{2}{5}\), cross - multiplying (which is valid as long as \(t-42
eq0\) and \(5
eq0\), and \(5
eq0\) is always true, and we will check \(t - 42
eq0\) at the end) gives us \(5t=- 2(t - 42)\).

Step2: Expand the right - hand side

Using the distributive property \(a(b + c)=ab+ac\) (here \(a=-2\), \(b = t\), \(c=-42\)), we get \(5t=-2t + 84\).

Step3: Add \(2t\) to both sides

Adding \(2t\) to both sides of the equation \(5t=-2t + 84\) to get all the \(t\) terms on one side. So \(5t+2t=-2t + 2t+84\), which simplifies to \(7t = 84\).

Step4: Solve for \(t\)

Divide both sides of the equation \(7t = 84\) by \(7\). \(\frac{7t}{7}=\frac{84}{7}\), so \(t = 12\).

Step5: Check the domain

We need to check if \(t = 12\) makes the denominator \(t - 42\) non - zero. When \(t = 12\), \(t-42=12 - 42=-30
eq0\), so \(t = 12\) is a valid solution.

Answer:

\(12\)